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<h1>Sizing of clock meshes</h1>
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<span class="comment">% Section 4, L. Vandenberghe, S. Boyd, and A. El Gamal</span>
<span class="comment">% "Optimal Wire and Transistor Sizing for Circuits with Non-Tree Topology"</span>
<span class="comment">% Original by Lieven Vanderberghe</span>
<span class="comment">% Adapted to CVX by Argyris Zymnis - 12/04/05</span>
<span class="comment">% Modified by Michael Grant - 3/8/06</span>
<span class="comment">%</span>
<span class="comment">% We consider the problem of sizing a clock mesh, so as to minimize the</span>
<span class="comment">% total dissipated power under a constraint on the dominant time constant.</span>
<span class="comment">% The numbers of nodes in the mesh is N per row or column (thus n=(N+1)^2</span>
<span class="comment">% in total). We divide the wire into m segments of width xi, i = 1,...,m</span>
<span class="comment">% which is constrained as 0 &lt;= xi &lt;= Wmax. We use a pi-model of each wire</span>
<span class="comment">% segment, with capacitance beta_i*xi and conductance alpha_i*xi.</span>
<span class="comment">% Defining C(x) = C0+x1*C1+x2*C2+...+xm*Cm we have that the dissipated</span>
<span class="comment">% power is equal to ones(1,n)*C(x)*ones(n,1). Thus to minimize the</span>
<span class="comment">% dissipated power subject to a constraint in the widths and a constraint</span>
<span class="comment">% in the dominant time constant, we solve the SDP</span>
<span class="comment">%               minimize        ones(1,m)*C(x)*ones(m,1)</span>
<span class="comment">%                   s.t.        Tmax*G(x) - C(x) &gt;= 0</span>
<span class="comment">%                               0 &lt;= xi &lt;= Wmax</span>

<span class="comment">%</span>
<span class="comment">% Circuit parameters</span>
<span class="comment">%</span>

dim=4;           <span class="comment">% grid is dimxdim (assume dim is even)</span>
n=(dim+1)^2;     <span class="comment">% number of nodes</span>
m=2*dim*(dim+1); <span class="comment">% number of wires</span>
                 <span class="comment">% 1...dim(dim+1) are horizontal segments</span>
                 <span class="comment">% (numbered rowwise);</span>
                 <span class="comment">% dim(dim+1)+1 ... 2*dim(dim+1) are vertical</span>
                 <span class="comment">% (numbered columnwise)</span>
beta = 0.5;      <span class="comment">% capacitance per segment is twice beta times xi</span>
alpha = 1;       <span class="comment">% conductance per segment is alpha times xi</span>
G0 = 1;          <span class="comment">% source conductance</span>
C0 = [ 10     2     7     5     3;
        8     3     9     5     5;
        1     8     4     9     3;
        7     3     6     8     2;
        5     2     1     9    10 ];
wmax = 1;       <span class="comment">% upper bound on x</span>

<span class="comment">%</span>
<span class="comment">% Build capacitance and conductance matrices</span>
<span class="comment">%</span>

CC = zeros(dim+1,dim+1,dim+1,dim+1,m+1);
GG = zeros(dim+1,dim+1,dim+1,dim+1,m+1);

<span class="comment">% constant term</span>
CC(:,:,:,:,1) = reshape( diag(C0(:)), dim+1, dim+1, dim+1, dim+1 );
zo13 = reshape( [1,0;0,1],   2, 1, 2, 1 );
zo24 = reshape( zo13,        1, 2, 1, 2 );
pn13 = reshape( [1,-1;-1,1], 2, 1, 2, 1 );
pn24 = reshape( pn13,        1, 2, 1, 2 );
<span class="keyword">for</span> i = 1 : dim+1,
    <span class="comment">% source conductance</span>
    <span class="comment">% first driver in the middle of row 1</span>
    GG(dim/2+1,i,dim/2+1,i,1) = G0;
    <span class="keyword">for</span> j = 1 : dim,
        <span class="comment">% horizontal segments</span>
        node = 1 + j + ( i - 1 ) * dim;
        CC([j,j+1],i,[j,j+1],i,node) = beta * zo13;
        GG([j,j+1],i,[j,j+1],i,node) = alpha * pn13;
        <span class="comment">% vertical segments</span>
        node = node + dim * ( dim + 1 );
        CC(i,[j,j+1],i,[j,j+1],node) = beta * zo24;
        GG(i,[j,j+1],i,[j,j+1],node) = alpha * pn24;
    <span class="keyword">end</span>
<span class="keyword">end</span>
<span class="comment">% reshape for ease of use in Matlab</span>
CC = reshape( CC, n*n, m+1 );
GG = reshape( GG, n*n, m+1 );

<span class="comment">%</span>
<span class="comment">% Compute points the tradeoff curve, and the three sample points</span>
<span class="comment">%</span>

npts    = 50;
delays  = linspace( 50, 150, npts );
xdelays = [ 50, 100 ];
xnpts   = length( xdelays );
areas   = zeros(1,npts);
<span class="keyword">for</span> i = 1 : npts  + xnpts,

    <span class="keyword">if</span> i &gt; npts,
        xi = i - npts;
        delay = xdelays(xi);
        disp( sprintf( <span class="string">'Particular solution %d of %d (Tmax = %g)'</span>, xi, xnpts, delay ) );
    <span class="keyword">else</span>,
        delay = delays(i);
        disp( sprintf( <span class="string">'Point %d of %d on the tradeoff curve (Tmax = %g)'</span>, i, npts, delay ) );
    <span class="keyword">end</span>

    <span class="comment">%</span>
    <span class="comment">% Construct and solve the convex model</span>
    <span class="comment">%</span>

    cvx_begin <span class="string">sdp</span> <span class="string">quiet</span>
        variable <span class="string">x(m)</span>
        variable <span class="string">G(n,n)</span> <span class="string">symmetric</span>
        variable <span class="string">C(n,n)</span> <span class="string">symmetric</span>
        dual <span class="string">variables</span> <span class="string">Y1</span> <span class="string">Y2</span> <span class="string">Y3</span> <span class="string">Y4</span> <span class="string">Y5</span>
        minimize( sum( C(:) ) )
        subject <span class="string">to</span>
            G == reshape( GG * [ 1 ; x ], n, n );
            C == reshape( CC * [ 1 ; x ], n, n );
            delay * G - C &gt;= 0;
            0 &lt;= x &lt;= wmax;
    cvx_end

    <span class="keyword">if</span> i &lt;= npts,
        areas(i) = sum(x);
    <span class="keyword">else</span>,
        xareas(xi) = sum(x);

        <span class="comment">%</span>
        <span class="comment">% Display sizes</span>
        <span class="comment">%</span>

        disp( sprintf( <span class="string">'Solution %d:'</span>, xi ) );
        disp( <span class="string">'Vertical segments:'</span> );
        reshape( x(1:dim*(dim+1),1), dim, dim+1 )
        disp( <span class="string">'Horizontal segments:'</span> );
        reshape( x(dim*(dim+1)+1:end), dim, dim+1 )

        <span class="comment">%</span>
        <span class="comment">% Determine the step responses</span>
        <span class="comment">%</span>

        figure(xi+1);
        A = -inv(C)*G;
        B = -A*ones(n,1);
        T = linspace(0,500,2000);
        Y = simple_step(A,B,T(2),length(T));
        indmax = 0;
        indmin = Inf;
        <span class="keyword">for</span> j = 1 : size(Y,1),
           inds = min(find(Y(j,:) &gt;= 0.5));
           <span class="keyword">if</span> ( inds &gt; indmax )
              indmax = inds;
              jmax = j;
           <span class="keyword">end</span>;
           <span class="keyword">if</span> ( inds &lt; indmin )
              indmin = inds;
              jmin = j;
           <span class="keyword">end</span>;
        <span class="keyword">end</span>;
        tthres = T(indmax);
        GinvC  = full( G \ C );
        tdom   = max(eig(GinvC));
        elmore = max(sum(GinvC'));
        hold <span class="string">off</span>; plot(T,Y(jmax,:),<span class="string">'-'</span>,T,Y(jmin,:));  hold <span class="string">on</span>;
        plot( tdom   * [1;1], [0;1], <span class="string">'--'</span>, <span class="keyword">...</span>
              elmore * [1;1], [0;1], <span class="string">'--'</span>, <span class="keyword">...</span>
              tthres * [1;1], [0;1], <span class="string">'--'</span>);
        axis([0 500 0 1])
        text(tdom,1,<span class="string">'d'</span>);
        text(elmore,1,<span class="string">'e'</span>);
        text(tthres,1,<span class="string">'t'</span>);
        text( T(600), Y(jmax,600), sprintf( <span class="string">'v%d'</span>, jmax ) );
        text( T(600), Y(jmin,600), sprintf( <span class="string">'v%d'</span>, jmin ) );
        title( sprintf( <span class="string">'Solution %d (Tmax=%g), fastest and slowest step responses'</span>, xi, delay ) );

    <span class="keyword">end</span>

<span class="keyword">end</span>;

<span class="comment">%</span>
<span class="comment">% Plot the tradeoff curve</span>
<span class="comment">%</span>

figure(1)
ind = isfinite(areas);
plot(areas(ind), delays(ind));
xlabel(<span class="string">'Area'</span>);
ylabel(<span class="string">'Tdom'</span>);
title(<span class="string">'Area-delay tradeoff curve'</span>);
hold <span class="string">on</span>
<span class="keyword">for</span> k = 1 : xnpts,
    text( xareas(k), xdelays(k), sprintf( <span class="string">'(%d)'</span>, k ) );
<span class="keyword">end</span>
</pre>
<a id="output"></a>
<pre class="codeoutput">
Point 1 of 50 on the tradeoff curve (Tmax = 50)
Point 2 of 50 on the tradeoff curve (Tmax = 52.0408)
Point 3 of 50 on the tradeoff curve (Tmax = 54.0816)
Point 4 of 50 on the tradeoff curve (Tmax = 56.1224)
Point 5 of 50 on the tradeoff curve (Tmax = 58.1633)
Point 6 of 50 on the tradeoff curve (Tmax = 60.2041)
Point 7 of 50 on the tradeoff curve (Tmax = 62.2449)
Point 8 of 50 on the tradeoff curve (Tmax = 64.2857)
Point 9 of 50 on the tradeoff curve (Tmax = 66.3265)
Point 10 of 50 on the tradeoff curve (Tmax = 68.3673)
Point 11 of 50 on the tradeoff curve (Tmax = 70.4082)
Point 12 of 50 on the tradeoff curve (Tmax = 72.449)
Point 13 of 50 on the tradeoff curve (Tmax = 74.4898)
Point 14 of 50 on the tradeoff curve (Tmax = 76.5306)
Point 15 of 50 on the tradeoff curve (Tmax = 78.5714)
Point 16 of 50 on the tradeoff curve (Tmax = 80.6122)
Point 17 of 50 on the tradeoff curve (Tmax = 82.6531)
Point 18 of 50 on the tradeoff curve (Tmax = 84.6939)
Point 19 of 50 on the tradeoff curve (Tmax = 86.7347)
Point 20 of 50 on the tradeoff curve (Tmax = 88.7755)
Point 21 of 50 on the tradeoff curve (Tmax = 90.8163)
Point 22 of 50 on the tradeoff curve (Tmax = 92.8571)
Point 23 of 50 on the tradeoff curve (Tmax = 94.898)
Point 24 of 50 on the tradeoff curve (Tmax = 96.9388)
Point 25 of 50 on the tradeoff curve (Tmax = 98.9796)
Point 26 of 50 on the tradeoff curve (Tmax = 101.02)
Point 27 of 50 on the tradeoff curve (Tmax = 103.061)
Point 28 of 50 on the tradeoff curve (Tmax = 105.102)
Point 29 of 50 on the tradeoff curve (Tmax = 107.143)
Point 30 of 50 on the tradeoff curve (Tmax = 109.184)
Point 31 of 50 on the tradeoff curve (Tmax = 111.224)
Point 32 of 50 on the tradeoff curve (Tmax = 113.265)
Point 33 of 50 on the tradeoff curve (Tmax = 115.306)
Point 34 of 50 on the tradeoff curve (Tmax = 117.347)
Point 35 of 50 on the tradeoff curve (Tmax = 119.388)
Point 36 of 50 on the tradeoff curve (Tmax = 121.429)
Point 37 of 50 on the tradeoff curve (Tmax = 123.469)
Point 38 of 50 on the tradeoff curve (Tmax = 125.51)
Point 39 of 50 on the tradeoff curve (Tmax = 127.551)
Point 40 of 50 on the tradeoff curve (Tmax = 129.592)
Point 41 of 50 on the tradeoff curve (Tmax = 131.633)
Point 42 of 50 on the tradeoff curve (Tmax = 133.673)
Point 43 of 50 on the tradeoff curve (Tmax = 135.714)
Point 44 of 50 on the tradeoff curve (Tmax = 137.755)
Point 45 of 50 on the tradeoff curve (Tmax = 139.796)
Point 46 of 50 on the tradeoff curve (Tmax = 141.837)
Point 47 of 50 on the tradeoff curve (Tmax = 143.878)
Point 48 of 50 on the tradeoff curve (Tmax = 145.918)
Point 49 of 50 on the tradeoff curve (Tmax = 147.959)
Point 50 of 50 on the tradeoff curve (Tmax = 150)
Particular solution 1 of 2 (Tmax = 50)
Solution 1:
Vertical segments:

ans =

    0.6555    0.4366    0.5234    0.4709    0.2363
    1.0000    0.8536    1.0000    0.9360    0.5700
    0.9233    0.2956    0.8004    1.0000    1.0000
    0.4130    0.1355    0.2672    0.6724    0.8867

Horizontal segments:

ans =

    0.1938    0.1436    0.0000    0.0000    0.0000
    0.0712    0.0633    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0945    0.1583
    0.0000    0.0000    0.0000    0.0865    0.0517

Particular solution 2 of 2 (Tmax = 100)
Solution 2:
Vertical segments:

ans =

    0.2688    0.0437    0.1712    0.1338    0.0736
    0.4135    0.0802    0.3064    0.2224    0.1485
    0.2576    0.0802    0.1120    0.3835    0.2816
    0.1344    0.0437    0.0245    0.2408    0.2453

Horizontal segments:

ans =

   1.0e-08 *

    0.2229    0.3083    0.2867    0.3101    0.2455
    0.1859    0.2203    0.2134    0.2305    0.1570
    0.2219    0.2200    0.2107    0.2218    0.1510
    0.2504    0.2411    0.2118    0.2290    0.2103

</pre>
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<img src="clock_mesh__01.png" alt=""> <img src="clock_mesh__02.png" alt=""> <img src="clock_mesh__03.png" alt=""> 
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